Entropy versus influence for complex functions of modulus one 2

TL;DR

This paper provides an example of a complex-valued function with low influence but high entropy of its Fourier coefficients, illustrating a separation between influence and entropy in Boolean functions.

Contribution

It introduces a specific function demonstrating that low influence does not necessarily imply low entropy of Fourier coefficients.

Findings

Function with influence ≤ 1 and entropy > (1/2) log n

Shows influence and entropy can be decoupled in complex functions

Highlights limitations of influence-based complexity measures

Abstract

This is a simplification of a previous version of this ArXiv note. We present an example of a function $f$ from $\{-1,1\}^n$ to the unit sphere in $\mathbb{C}$ with influence bounded by $1$ and entropy of $|\hat f|^2$ larger than $\frac12\log n$.